Approximating the moments of marginals of high-dimensional distributions

For probability distributions on $\mathbb{R}^n$, we study the optimal sample size N = N(n,p) that suffices to uniformly approximate the pth moments of all one-dimensional marginals. Under the assumption that the marginals have bounded 4p moments, we obtain the optimal bound $N=O(n^{p/2})$ for p > 2. This bound goes in the direction of bridging the two recent results: a theorem of Guedon and Rudelson [Adv. Math. 208 (2007) 798-823] which has an extra logarithmic factor in the sample size, and a result of Adamczak et al. [J. Amer. Math. Soc. 23 (2010) 535-561] which requires stronger subexponential moment assumptions.Comment: Published in at http://dx.doi.org/10.1214/10-AOP589 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Concentration inequalities for random tensors

We show how to extend several basic concentration inequalities for simple random tensors $X = x_1 \otimes \cdots \otimes x_d$ where all $x_k$ are independent random vectors in $\mathbb{R}^n$ with independent coefficients. The new results have optimal dependence on the dimension $n$ and the degree $d$. As an application, we show that random tensors are well conditioned: $(1-o(1)) n^d$ independent copies of the simple random tensor $X \in \mathbb{R}^{n^d}$ are far from being linearly dependent with high probability. We prove this fact for any degree $d = o(\sqrt{n/\log n})$ and conjecture that it is true for any $d = O(n)$.Comment: A few more typos were correcte

Four lectures on probabilistic methods for data science

Methods of high-dimensional probability play a central role in applications for statistics, signal processing theoretical computer science and related fields. These lectures present a sample of particularly useful tools of high-dimensional probability, focusing on the classical and matrix Bernstein's inequality and the uniform matrix deviation inequality. We illustrate these tools with applications for dimension reduction, network analysis, covariance estimation, matrix completion and sparse signal recovery. The lectures are geared towards beginning graduate students who have taken a rigorous course in probability but may not have any experience in data science applications.Comment: Lectures given at 2016 PCMI Graduate Summer School in Mathematics of Data. Some typos, inaccuracies fixe

Frame expansions with erasures: an approach through the non-commutative operator theory

In modern communication systems such as the Internet, random losses of information can be mitigated by oversampling the source. This is equivalent to expanding the source using overcomplete systems of vectors (frames), as opposed to the traditional basis expansions. Dependencies among the coefficients in frame expansions often allow for better performance comparing to bases under random losses of coefficients. We show that for any n-dimensional frame, any source can be linearly reconstructed from only (n log n) randomly chosen frame coefficients, with a small error and with high probability. Thus every frame expansion withstands random losses better (for worst case sources) than the orthogonal basis expansion, for which the (n log n) bound is attained. The proof reduces to M.Rudelson's selection theorem on random vectors in the isotropic position, which is based on the non-commutative Khinchine's inequality.Comment: 12 page

Integer cells in convex sets

Every convex body K in R^n has a coordinate projection PK that contains at least vol(0.1 K) cells of the integer lattice PZ^n, provided this volume is at least one. Our proof of this counterpart of Minkowski's theorem is based on an extension of the combinatorial density theorem of Sauer, Shelah and Vapnik-Chervonenkis to Z^n. This leads to a new approach to sections of convex bodies. In particular, fundamental results of the asymptotic convex geometry such as the Volume Ratio Theorem and Milman's duality of the diameters admit natural versions for coordinate sections.Comment: Historical remarks on the notion of the combinatorial dimension are added. This is a published version in Advances in Mathematic

Isoperimetry of waists and local versus global asymptotic convex geometries

Existence of nicely bounded sections of two symmetric convex bodies K and L implies that the intersection of random rotations of K and L is nicely bounded. For L = subspace, this main result immediately yields the unexpected phenomenon: "If K has one nicely bounded section, then most sections of K are nicely bounded". This 'existence implies randomness' consequence was proved independently in [Giannopoulos, Milman and Tsolomitis]. The main result represents a new connection between the local asymptotic convex geometry (study of sections of convex bodies) and the global asymptotic convex geometry (study of convex bodies as a whole). The method relies on the new 'isoperimetry of waists' on the sphere due to Gromov